R0053-27111 · q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0086
完整finite laws與total precommitted theta-independent interval procedure責任。操作=ci-interpretation-boundary；資料={"mode":"ignored_selection","model":{"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[11,13],[2,13]],[[2,13],[11,13]]],"intervals":[[0,0],[1,1]],"selected_samples":["a"]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。
G001 — 完整finite laws與total precommitted theta-independent interval procedure責任。操作=ci-interpretation-boundary；資料={"mode":"ignored_selection","model":{"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[11,13],[2,13]],[[2,13],[11,13]]],"intervals":[[0,0],[1,1]],"selected_samples":["a"]}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。
操作前
source roles = [ci-interpretation-boundary]
操作後
derived = [[mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary]]
條件：無
此頁直接列完整來源欄序與mode特定角色，不靠JavaScript或由答案外形猜介面；No、undefined、infinity各保原意。
規則：source_ordered_fields_without_JS · 依賴：G001
操作前
source roles = [[0, 1], [a, b], [[[11, 13], [2, 13]], [[2, 13], [11, 13]]], [[0, 0], [1, 1]]]
操作後
derived = [[0, 1], [a, b], [[0, 0], [1, 1]], [[11/13, 2/13], [2/13, 11/13]], [[a, [0, 0], 0], [b, [1, 1], 1]]]
條件：無
interval先綁原始samples，機率row先綁原始theta與原始sample欄，再連資料一起排序；Source13 theta用numeric，samples用English canonical。區間閉合且有序，每個theta law非負正規化。同一預先固定、total的I(X)不依未知theta。
規則：CI_ORIGINAL_theta_sample_interval_probability_alignment_Source13_numeric · 依賴：G001, S001
操作前
CI theta0 ORIGINAL probability normalization column0 := sum(0, 11/13)
操作後
CI theta0 ORIGINAL probability normalization column0 = 11/13
條件：無
逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。
規則：exact_rational_arithmetic · 依賴：G001, S002
操作前
CI theta0 ORIGINAL probability normalization column1 := sum(11/13, 2/13)
操作後
CI theta0 ORIGINAL probability normalization column1 = 1
條件：無
逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。
規則：exact_rational_arithmetic · 依賴：G001, S003
操作前
CI theta1 ORIGINAL probability normalization column0 := sum(0, 2/13)
操作後
CI theta1 ORIGINAL probability normalization column0 = 2/13
條件：無
逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。
規則：exact_rational_arithmetic · 依賴：G001, S004
操作前
CI theta1 ORIGINAL probability normalization column1 := sum(2/13, 11/13)
操作後
CI theta1 ORIGINAL probability normalization column1 = 1
條件：無
逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。
規則：exact_rational_arithmetic · 依賴：G001, S005
操作前
CI theta0 ORIGINAL included coverage column0 := sum(0, 11/13)
操作後
CI theta0 ORIGINAL included coverage column0 = 11/13
條件：無
逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。
規則：exact_rational_arithmetic · 依賴：G001, S006
操作前
source roles = [0, [a, b], [[0, 0], [1, 1]], [11/13, 2/13]]
操作後
derived = [[[a, 0, 0, 0, 是, 11/13], [b, 1, 1, 0, 否, 2/13]], 11/13]
條件：無
固定真theta，逐所有可能samples判L(x)<=theta<=U(x)，兩端相等也算hit，只把hit的P_theta(x)加起。這是反覆抽樣coverage，不是觀測資料後theta的posterior。
規則：CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · 依賴：G001, S007
操作前
CI theta1 ORIGINAL included coverage column1 := sum(0, 11/13)
操作後
CI theta1 ORIGINAL included coverage column1 = 11/13
條件：無
逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。
規則：exact_rational_arithmetic · 依賴：G001, S008
操作前
source roles = [1, [a, b], [[0, 0], [1, 1]], [2/13, 11/13]]
操作後
derived = [[[a, 0, 0, 1, 否, 2/13], [b, 1, 1, 1, 是, 11/13]], 11/13]
條件：無
固定真theta，逐所有可能samples判L(x)<=theta<=U(x)，兩端相等也算hit，只把hit的P_theta(x)加起。這是反覆抽樣coverage，不是觀測資料後theta的posterior。
規則：CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · 依賴：G001, S009
操作前
CI theta0 ORIGINAL selected mass column0 := sum(0, 11/13)
操作後
CI theta0 ORIGINAL selected mass column0 = 11/13
條件：無
逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。
規則：exact_rational_arithmetic · 依賴：G001, S010
操作前
CI theta0 ORIGINAL selected hit column0 := sum(0, 11/13)
操作後
CI theta0 ORIGINAL selected hit column0 = 11/13
條件：無
逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。
規則：exact_rational_arithmetic · 依賴：G001, S011
操作前
CI theta0 ORIGINAL selected conditional coverage := div(11/13, 11/13)
操作後
CI theta0 ORIGINAL selected conditional coverage = 1
條件：無
逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。
規則：exact_rational_arithmetic · 依賴：G001, S012
操作前
source roles = [0, [a], 11/13, 11/13]
操作後
derived = [1]
條件：無
先計selection機率mass，再在mass>0時計hit/mass；mass0保留literal undefined，不能寫0或作0/0。selection納入procedure後可合法，錯誤是未重算就沿用coverage，不是宣稱所有selection必無效。
規則：CI_EACH_theta_selected_MASS_hit_and_undefined_at_zero · 依賴：G001, S013
操作前
CI theta1 ORIGINAL selected mass column0 := sum(0, 2/13)
操作後
CI theta1 ORIGINAL selected mass column0 = 2/13
條件：無
逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。
規則：exact_rational_arithmetic · 依賴：G001, S014
操作前
CI theta1 ORIGINAL selected conditional coverage := div(0, 2/13)
操作後
CI theta1 ORIGINAL selected conditional coverage = 0
條件：無
逐原始角色與已證中間量精確代值；不以近似值、索引位址或自洽但換過的操作數取代來源。
規則：exact_rational_arithmetic · 依賴：G001, S015
操作前
source roles = [1, [a], 2/13, 0]
操作後
derived = [0]
條件：無
先計selection機率mass，再在mass>0時計hit/mass；mass0保留literal undefined，不能寫0或作0/0。selection納入procedure後可合法，錯誤是未重算就沿用coverage，不是宣稱所有selection必無效。
規則：CI_EACH_theta_selected_MASS_hit_and_undefined_at_zero · 依賴：G001, S016
操作前
source roles = [[a], [[0, 11/13], [1, 11/13]]]
操作後
derived = [[a], [[0, 11/13, 1], [1, 2/13, 0]], 否]
條件：無
來源把selected samples當集合去重並canonical排序，合法duplicate不當成新sample或自動錯誤；條件coverage逐theta獨立重算，可能與原coverage巧合相同。
規則：CI_selected_samples_extensional_dedup_not_automatic_invalidity · 依賴：G001, S017
S018 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。S017 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。S016 · premise_and_result_audit — 以精確有理數重算 CI theta1 ORIGINAL selected conditional coverage，不以小數近似替代。S015 · premise_and_result_audit — 以精確有理數重算 CI theta1 ORIGINAL selected mass column0，不以小數近似替代。S014 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。S013 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL selected conditional coverage，不以小數近似替代。S012 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL selected hit column0，不以小數近似替代。S011 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL selected mass column0，不以小數近似替代。S010 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。S009 · premise_and_result_audit — 以精確有理數重算 CI theta1 ORIGINAL included coverage column1，不以小數近似替代。S008 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。S007 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL included coverage column0，不以小數近似替代。S006 · premise_and_result_audit — 以精確有理數重算 CI theta1 ORIGINAL probability normalization column1，不以小數近似替代。S005 · premise_and_result_audit — 以精確有理數重算 CI theta1 ORIGINAL probability normalization column0，不以小數近似替代。S004 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL probability normalization column1，不以小數近似替代。S003 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL probability normalization column0，不以小數近似替代。S002 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。S001 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。(ignored_selection, [a], [[0, 11/13], [1, 11/13]], [[0, 11/13, 1], [1, 2/13, 0]], 否, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)
依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。
覆核日期：2026-10-26
原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。
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